Non-diagonalizable and non-divergent susceptibility tensor in the Hamiltonian mean-field model with asymmetric momentum distributions

TL;DR

This paper studies the susceptibility tensor in the Hamiltonian mean-field model with asymmetric momentum distributions, revealing it can be non-diagonalizable and non-divergent, with theoretical predictions confirmed by numerical simulations.

Contribution

It demonstrates that the susceptibility tensor in the model can be asymmetric and non-diagonalizable for asymmetric distributions, a novel insight into long-range interacting systems.

Findings

Susceptibility tensor is non-symmetric and non-diagonalizable.

Tensor remains finite at the stability threshold.

Numerical simulations confirm theoretical predictions.

Abstract

We investigate response to an external magnetic field in the Hamiltonian mean-field model, which is a paradigmatic toy model of a ferromagnetic body and consists of plane rotators like the XY spins. Due to long-range interactions, the external field drives the system to a long-lasting quasistationary state before reaching thermal equilibrium, and the susceptibility tensor obtained in the quasista- tionary state is predicted by a linear response theory based on the Vlasov equation. For spatially homogeneous stable states, whose momentum distributions are asymmetric with zero-means, the theory reveals that the susceptibility tensor for an asymptotically constant external field is neither symmetric nor diagonalizable, and the predicted states are not stationary accordingly. Moreover, the tensor has no divergence even at the stability threshold. These theoretical findings are confirmed by direct numerical simulations of the Vlasov equation for the skew-normal distribution functions.